Theorem eqelssd 3166

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 114	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3153	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3164	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134

This theorem is referenced by:  fiuni  6955  ennnfonelemrn  12374  ennnfonelemdm  12375  unirnblps  13216  unirnbl  13217  dvidlemap  13454  dviaddf  13463  dvimulf  13464  dvcj  13467  dvrecap  13471